If the angle between a pair of tangents drawn | vedclass

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(D) The given circle is $x^2+y^2-4x+2y+3=0$. Comparing with $x^2+y^2+2gx+2fy+c=0$,we get $g=-2, f=1, c=3$. The center is $(2, -1)$ and the radius $r =

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$x^2+y^2-4x+2y+1=0$

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(D) The given circle is $x^2+y^2-4x+2y+3=0$. Comparing with $x^2+y^2+2gx+2fy+c=0$,we get $g=-2, f=1, c=3$. The center is $(2, -1)$ and the radius $r = \sqrt{g^2+f^2-c} = \sqrt{4+1-3} = \sqrt{2}$. If the angle between the tangents is $\frac{\pi}{2}$,the locus of the point $P$ is the director circle. The equation of the director circle is $(x-h)^2+(y-k)^2 = 2r^2$,where $(h, k)$ is the center. Substituting the values: $(x-2)^2+(y+1)^2 = 2(\sqrt{2})^2 = 4$. Expanding this: $x^2-4x+4+y^2+2y+1 = 4$. $x^2+y^2-4x+2y+1 = 0$. Thus,option $D$ is correct.

If the angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2-4x+2y+3=0$ is $\frac{\pi}{2}$,then the locus of $P$ is

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